Based on Gödel's work, I don't think you can actually get to that state.
Speaking as a very,
very mediocre formal logician, I'm inclined to sorta-kinda-ish agree.
Going off Gödel's two incompleteness theorems, my understanding is that you couldn't ever create, say, a completely axiomatised and self-proving theory of mathematics, to steal the nice phrasing Jev used. It'd be a (clearly non-trivial) formal system, but therein there would exist statements that couldn't be proved in that system -- and, to bring in the second theorem, if that system can be proved to be consistent using its own logic, then it must itself be inconsistent.
This can, I assume, be extended to physics, to direct this post towards a field with which I'm more familiar: a 'theory of everything' to fully explain and link all physical phenomena capable of predicting the outcome of any experiment that could possibly be carried out (in principle) would qualify quite certainly as a non-trivial, consistent mathematical system; a TOE wouldn't necessarily be
impossible, but to be able to rigorously define it as a final theory would be a different matter.
Something that struck me midway through writing this was that, if one of the implications of Gödel's incompleteness theorems is that (pure) mathematics is inexhaustible, then so too must physics as a finite superset of the laws of mathematics, i.e. there would always be new problems that couldn't be solved within existing rules. How this would tie in with the SoCT, I'm uncertain, due in no small part to my general fuzziness with this field -- would the existence of new problems as a result of the incompleteness theorems keep them going? Would they reach Istvaan's outcome as a result of being aware of the two theorems alongside everything else they know? Something else entirely?